Understanding Tangent Lines and Linear Approximations

The video tutorial explains how to find the equation of a tangent line to a function at a given point and use it for linear approximation. It covers determining the point of tangency, calculating the slope using derivatives, and formulating the tangent line equation. The tutorial also demonstrates how to use the tangent line for approximating function values and calculating the error between the approximation and the actual function value.

What is the main goal of the problem introduced in the video?

• To solve the function for x=0

• To determine the equation of the tangent line at x=1

• To find the maximum value of the function

• To calculate the integral of the function

What is the x-coordinate of the point of tangency?

• 1

• 2

• 0

• 3

Which rule is applied to find the derivative of the function?

• Product Rule

• Power Rule

• Chain Rule

• Quotient Rule

What is the slope of the tangent line at x=1?

• 2

• 0

• 1

• -2

What form is used to find the equation of the tangent line?

• Standard Form

• Point-Slope Form

• Vertex Form

• Slope-Intercept Form

What is the equation of the tangent line in slope-intercept form?

• y = 2x - 5

• y = 2x + 5

• y = -2x + 5

• y = -2x - 5

What is the linear approximation for f(1.1)?

• 2.5

• 2.6

• 2.8

• 2.7

What is the actual value of f(1.1) calculated using the original function?

• 2.96076

• 2.86076

• 3.06076

• 2.76076

What is the error between the linear approximation and the actual function value?

• 0.04924

• 0.03924

• 0.05924

• 0.02924

Why is the error considered positive?

• Because the function is increasing

• Because the approximation is always larger

• Because the approximation is always smaller

• Because absolute value is taken